Periodica Mathematica Hungarica Vol. 51 (2), 2005, pp. 15–18
A NOTE ON THE SIZE OF THE LARGEST BALL
INSIDE A CONVEX POLYTOPE
Imre Bárány (Budapest, London) and Nándor Simányi (Birmingham)
[Communicated by: Gábor Tardos ]
Abstract
Let m > 1 be an integer, Bm the set of all unit vectors of Rm pointing in the
direction of a nonzero integer vector of the cube [−1, 1]m . Denote by sm the radius
of the largest ball contained in the convex hull of Bm . We determine the exact value
2
.
of sm and obtain the asymptotic equality sm ∼ √log
m
1. Introduction
Let m ≥ 2 be an integer, and consider the sets
v m
v ∈ Am .
Am = {−1, 0, 1} \ 0 , and Bm =
v Let Cm be the convex hull of Bm , and sm the radius of the largest ball contained
in Cm . (Due to the apparent symmetries of Cm , such a largest ball is necessarily
centered at the origin.) In the paper [B-M-S] (dealing with rotation numbers/vectors
of billiards) we needed sharp lower and upper estimates for the extremal radius sm .
Here we determine the exact value of sm which, of course, implies such estimates.
Theorem.


sm = 
m
k=1
−1/2
1

√
2 
√
k+ k−1
.
The Theorem implies that
5
1
1
log m < s−2
log m + ,
m <
4
4
4
where log denotes the natural logarithm. As an immediate corollary, the quantity
2
sm is asymptotically equal to √log
.
m
Mathematics subject classification number: 52B11, Secondary 52B12.
Key words and phrases: convex polytopes, inscribed ball, baricentric subdivision.
0031-5303/2005/$20.00
c Akadémiai Kiadó, Budapest
Akadémiai Kiadó, Budapest
Springer, Dordrecht
16
i. bárány and n. simányi
2. Proof of the Theorem
The proof will be split into a few lemmas. The first one of them is a trivial
observation.
Lemma 1. The set of vertices Bm of the convex polytope Cm , and hence Cm
itself, is invariant under the action of the full isometry group G of the cube [−1, 1]m .
(The group G is generated by all permutations of the coordinates in Rm , and by all
reflections across the coordinate hyperplanes.)
k
We will use the notation vk = √1k i=1 ei (k = 1, . . . , m) for some specific
vertices of Cm . (Here ei stands for the i-th standard unit vector of Rm .)
Lemma 2. The simplex S, spanned by the linearly independent vectors vk
(k = 1, . . . , m) as vertices, is a face of the polytope
√
√Cm whose outer normal vector
is u = (u1 , . . . , um ) with the coordinates ui = i − i − 1.
Proof. Consider the scalar product function v, u (v ∈ Bm ) restricted to
the set Bm of vertices of the polytope Cm . Elementary inspection shows that this
scalar product function can only attain its maximum value at the vertices vk , and
actually,
vk , u = 1
for each k = 1, . . . , m. This proves all claims of the lemma.
(1)
Lemma 3. For any face F of the polytope Cm there exists a congruence g ∈ G
such that g(F ) = S.
Proof. Fix a non-zero vector w = (w1 , . . . , wm ) whose ray R(w) = {λw |
λ ≥ 0} intersects the interior of the face F . By selecting w in a generic manner,
we can assume that the absolute values |wi | of its coordinates are distinct and all
different from zero. Therefore, by applying a suitable element g ∈ G, we can even
assume that
w1 > w2 > · · · > wm > 0.
(2)
We claim that g(F ) = S. Indeed, by (2) we have the linear expansion
w=
m √
k (wk − wk+1 )vk
k=1
of w in the basis {v1 , . . . , vm } with positive coefficients. (With the natural convention wm+1 = 0.) This proves that some positive multiple of w is a convex linear
combination of the vertices of S with non-zero coefficients, so the face g(F ) shares
an interior point with S.
a note on the size of the largest ball inside a convex polytope
17
It follows from the previous lemma that the radius sm of the inscribed sphere
is actually the distance between S and the origin. However, this distance is equal
to sm = u, e1 /u = 1/u by (1). It is clear that
u2 =
m
k=1
1
√
2 ,
√
k+ k−1
finishing the proof of our theorem.
m
Define Rm = k=1 k1 . For the asymptotic value of sm we use the elementary
fact that log m < Rm < log m + 1.
m
m
k=1
k=1
1
1
log m <
<
√
4
4k
<1+
m
k=2
1
2
2 = u
√
k+ k−1
1
1
< 1 + (log m + 1)
4(k − 1)
4
1
5
= log m + .
4
4
Remark 1. Let K be the convex cone generated by the vectors vk , k =
1, . . . , m. The meaning of Lemma 3 is that the cones g(K) (g ∈ G) form a triangulation of the space Rm . As a matter of fact, the intersections of the cones g(K)
with the standard (m − 1)-simplex
Sm−1 =
x∈R
m
m
xi = 1, xi ≥ 0 for all i
i=1
form the baricentric subdivision of Sm−1 .
Remark 2. The following natural question has been considered in several
papers, for instance in [B-F] and [B-W]. What is the maximal radius r(m, N ) of the
inscribed ball of the convex hull of N points chosen from the unit ball of Rm ? In
our case N = 3m − 1 and one may wonder how close sm and Bm are to the maximal
radius and best arrangement. It turns out that they are very far: it follows from
the results of [B-F], [R], and [B-W] that, in the given range N = 3m − 1,
1/2
8
(1 + o(1))
r(m, N ) =
9
as m → ∞. So the optimal radius is much larger than sm . This also shows that, as
expected, Bm is far from being distributed uniformly on the unit sphere.
18
i. bárány and n. simányi
Acknowledgement. The authors express their sincere gratitude to Michal
Misiurewicz (Indiana University Purdue University at Indianapolis) for posing the
above problem during his joint research with the second author and Alexander Blokh
(University of Alabama at Birmingham). The first named author was partially
supported by Hungarian National Foundation Grants T 037846 and T 046246, and
the second author received partial support from the NSF grant DMS 0457168.
References
[B-F]
I. Bárány and Z. Füredi, Approximation of the sphere by polytopes having few
vertices, Proc. AMS 102 (1988), 651–659.
[B-M-S] A. Blokh, M. Misiurewicz and N. Simányi, Rotation sets of billiards with one
obstacle, Manuscript, 2005.
[B-W] K. Böröczky Jr. and G. Wintsche, Covering the sphere by equal spherical balls,
in: Discrete and computational geometry: the Goodman–Pollach Festschrift (ed. by
S. Basu et al.), 2003, 237–253.
[R]
C. A. Rogers, Covering a sphere with spheres Mathematika 10 (1963), 157–164.
(Received: June 21, 2005)
Imre Bárány
Alfréd Rényi Mathematical Institute
Hungarian Academy of Sciences
H-1364 Budapest, P.O. Box 127
Hungary
and
Department of Mathematics
University College London
Gower Street, London, WC1E 6BT
United Kingdom
E-mail: [email protected]
Nándor Simányi
University of Alabama at Birmingham
Department of Mathematics
Campbell Hall, Birmingham, AL 35294
U.S.A.
E-mail: [email protected]